Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $t \neq 0$. $q = \dfrac{6(3t - 10)}{8t} \div \dfrac{9t - 30}{-5} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $q = \dfrac{6(3t - 10)}{8t} \times \dfrac{-5}{9t - 30} $ When multiplying fractions, we multiply the numerators and the denominators. $q = \dfrac{ 6(3t - 10) \times -5 } { 8t \times (9t - 30) } $ $ q = \dfrac {-5 \times 6(3t - 10)} {8t \times 3(3t - 10)} $ $ q = \dfrac{-30(3t - 10)}{24t(3t - 10)} $ We can cancel the $3t - 10$ so long as $3t - 10 \neq 0$ Therefore $t \neq \dfrac{10}{3}$ $q = \dfrac{-30 \cancel{(3t - 10})}{24t \cancel{(3t - 10)}} = -\dfrac{30}{24t} = -\dfrac{5}{4t} $